Repeated and final commutators in group actions. (Q2846732)

Let \(G\) be a finite group, \(A\ltimes G\) a semidirect product. The repeated commutators are the subgroups \(K=[G,A,\ldots,A]\) of \(G\), the smallest one is called the final commutator. By P. Hall's classical result, if the final commutator is the trivial group then both \([G,A]\) and \(A/C_G(A)\) are nilpotent.NEWLINENEWLINE The authors extend Hall's results omitting the hypothesis that the final commutator be trivial. \(K\) is \(A\)-invariant and hence \(A\) permutes the left cosets of \(K\) in \(G\), denote this permutation group by \(P(G,A,K)\). Let \(\mathcal X\) be a class of groups which is closed under taking subgroups, homomorphic images and products of normal subgroups, and denote by \(G^{\mathcal X}\) the minimal normal subgroup of \(G\) such that its factor is in \(\mathcal X\). The main results of the paper are the following. If \([G,A]^{\mathcal X}\subseteq K\) then \(P(G,A,K)\) is an \(\mathcal X\)-group. Moreover, the final commutator \(K\) is normal if and only if the final term of the lower central series of \([G,A]\) is contained in \(K\).